Stability Polynomial

Consider the general linear multistep method

(8.26)

which we assume to be consistent and zero–stable. The theoretical solution of the initial value problem satisfies

(8.27)

where , the local truncation error. If we denote by the solution of (8.26) when a round-off error is committed at the application of the method, then

(8.28)

on subtracting (8.28) from (8.27) and defining the global error by we find

(8.29)

where

If we assume that the partial derivatives exists for all t , then by the mean value theorem, there exists a number lying in the open interval whose end points are and , such that